Sampling from distributions play a crucial role in aiding practitioners with statistical inference. However, in numerous situations, obtaining exact samples from complex distributions is infeasible. Consequently, researchers often turn to approximate sampling techniques to address this challenge. Fast approximate sampling from complicated distributions has gained much traction in the last few years with considerable progress in this field. Previous work has shown that for some problems a preconditioning can make the algorithm faster. In our research, we explore the Langevin Monte Carlo (LMC) algorithm and demonstrate its effectiveness in enabling inference from the obtained samples. Additionally, we establish a convergence rate for the LMC Markov chain in total variation. Lastly, we derive non-asymptotic bounds for approximate sampling from specific target distributions in the Wasserstein distance, particularly when the preconditioning is spatially invariant.

翻译：从分布中采样在帮助实践者进行统计推断方面起着关键作用。然而，在许多情况下，从复杂分布中获取精确样本是难以实现的。因此，研究人员常转向近似采样技术来应对这一挑战。近年来，从复杂分布中进行快速近似采样备受关注，该领域取得了显著进展。已有研究表明，对于某些问题，使用预条件处理可以加速算法运行。在我们的研究中，我们探索了朗之万蒙特卡洛（LMC）算法，并证明了其在从所得样本中进行推断的有效性。此外，我们建立了LMC马尔可夫链在总变差距离下的收敛速率。最后，我们推导了在Wasserstein距离下，特别是当预条件处理为空间不变时，从特定目标分布进行近似采样的非渐近界。